A competitive pharmaceutical supply chain under the marketing mix strategies and product life cycle with a fuzzy stochastic demand

Abstract

This paper addresses a competition problem in two Pharmaceutical Supply Chains (PSCs) with their exclusive retailer and manufacturer. However, the key to successful manufacturing is not just understanding a Product Life Cycle (PLC), but also proactively managing products throughout their lifetime, applying the appropriate resources and sales and marketing strategies, depending on what stage products are at in the cycle. Providing a competitive pharmaceutical supply chain under a product life cycle and offering Nash and Stackelberg games under the marketing mix strategies (i.e., price, product quality, location, and advertising) and product life cycle (i.e., introduction, growth, and maturity) is a novel idea that needs to be studied. So, a demand function depends on the marketing mix strategies. The competition between PSCs is considered based on the levels of price, quality, access, and promotion (i.e., marketing mix) across different stages of their PLC. Concerning the PLC and the position of a supply chain in a specific period of their PLC, the considered competitions are Nash and Stackelberg. The results of the level of the marketing mix, demand, and profit for each supply chain are obtained by given different games in line with the stage of their PLC. Furthermore, based on investigating the previous annual data, the sensitivity coefficients of the marketing mix of PSCs in the demand function, these coefficients are considered as stochastic fuzzy parameters. Based on the obtained results, the average total profit of the two supply chains in the product life cycle, at Nash equilibrium, is 6.5 times that of Stackelberg. The average total profit of the first supply chain in the product life cycle, at Stackelberg, is 10% higher than Nash equilibrium. Moreover, the average total profit of the second supply chain in the product life cycle, at Nash equilibrium, is 19 times that of Stackelberg.

Appendices

Appendix 1: Nomenclature

Indices and parameters of the equations are shown before Section 1.

Appendix 2

$G_{2} = c_{1} + c^{\prime}_{1} - b_{1} \tau_{1} - b^{\prime}_{1} \tau_{1}$	$G_{1} = c_{1} + c^{\prime}_{1} + b_{1} \tau_{1} + b^{\prime}_{1} \tau_{1}$
$G_{4} = d_{1} + d^{\prime}_{1} - b_{1} \tau^{\prime}_{1} - b^{\prime}_{1} \tau^{\prime}_{1}$	$G_{3} = d_{1} + d^{\prime}_{1} + b_{1} \tau^{\prime}_{1} + b^{\prime}_{1} \tau^{\prime}_{1}$
$G_{6} = e_{1} + e^{\prime}_{1} - b_{1} \tau^{\prime\prime}_{1} - b^{\prime}_{1} \tau^{\prime\prime}_{1}$	$G_{5} = e_{1} + e^{\prime}_{1} + b_{1} \tau^{\prime\prime}_{1} + b^{\prime}_{1} \tau^{\prime\prime}_{1}$
$G_{8} = c_{2} + c^{\prime}_{2} - b_{2} \tau_{2} - b^{\prime}_{2} \tau_{2}$	$G_{7} = c_{2} + c^{\prime}_{2} + b_{2} \tau_{2} + b^{\prime}_{2} \tau_{2}$
$G_{10} = d_{2} + d^{\prime}_{2} - b_{2} \tau^{\prime}_{2} - b^{\prime}_{2} \tau^{\prime}_{2}$	$G_{9} = d_{2} + d^{\prime}_{2} + b_{2} \tau^{\prime}_{2} + b^{\prime}_{2} \tau^{\prime}_{2}$
$G_{12} = e_{2} + e^{\prime}_{2} - b_{2} \tau^{\prime\prime}_{2} - b^{\prime}_{2} \tau^{\prime\prime}_{2}$	$G_{11} = e_{2} + e^{\prime}_{2} + b_{2} \tau^{\prime\prime}_{2} + b^{\prime}_{2} \tau^{\prime\prime}_{2}$
${\text{G}}_{13} = G_{1} G_{2} \varepsilon_{q1} \varepsilon_{a1} + G_{3} G_{4} \varepsilon_{s1} \varepsilon_{a1} + G_{5} G_{6} \varepsilon_{s1} \varepsilon_{q1}$
${\text{G}}_{14} = G_{2} \tau_{1} \varepsilon_{q1} \varepsilon_{a1} + \varepsilon_{s1} \varepsilon_{q1} \varepsilon_{a1} + G_{4} \tau_{1}^{\prime } \varepsilon_{s1} \varepsilon_{a1} + G_{6} \tau_{1}^{\prime \prime } \varepsilon_{s1} \varepsilon_{q1}$
${\text{G}}_{15} = G_{8} c^{\prime}_{1} \varepsilon_{q2} \varepsilon_{a2} + G_{10} d^{\prime}_{1} \varepsilon_{s2} \varepsilon_{a2} + G_{12} e^{\prime}_{1} \varepsilon_{s2} \varepsilon_{a2}$
${\text{G}}_{16} = G_{8} \tau_{2} \varepsilon_{q2} \varepsilon_{a2} + \varepsilon_{s2} \varepsilon_{q2} \varepsilon_{a2} + G_{10} \tau_{2}^{\prime } \varepsilon_{s2} \varepsilon_{a2} + G_{12} \tau_{2}^{\prime \prime } \varepsilon_{s2} \varepsilon_{q2}$
${\text{G}}_{17} = G_{7} G_{8} \varepsilon_{q2} \varepsilon_{a2} + G_{9} G_{10} \varepsilon_{s2} \varepsilon_{a2} + G_{11} G_{12} \varepsilon_{s2} \varepsilon_{q2}$
${\text{G}}_{18} = G_{2} c^{\prime}_{2} \varepsilon_{q2} \varepsilon_{a2} + G_{4} d^{\prime}_{2} \varepsilon_{s2} \varepsilon_{a2} + G_{6} e^{\prime}_{2} \varepsilon_{s2} \varepsilon_{a2}$
${\text{G}}_{19} = \frac{1}{{2G_{14} (b_{1} + b_{1}^{\prime } ) + G_{13} }}$
${\text{G}}_{20} = b^{\prime}_{2} c_{1}^{\prime } G_{14} G_{19} + c^{\prime}_{2} \varepsilon_{q1} \varepsilon_{a1} c_{1}^{\prime } G_{2} G_{19} + d^{\prime}_{2} \varepsilon_{s1} \varepsilon_{a1} c_{1}^{\prime } G_{4} G_{19} + e^{\prime}_{2} \varepsilon_{s1} \varepsilon_{q1} c_{1}^{\prime } G_{6} G_{19}$
$G_{21} = - c^{\prime}_{2} \varepsilon_{q1} \varepsilon_{a1} G_{2} G_{19} - d^{\prime}_{2} \varepsilon_{s1} \varepsilon_{a1} G_{4} G_{19} - e^{\prime}_{2} \varepsilon_{s1} \varepsilon_{q1} G_{6} G_{19}$
$G_{22} = - b_{2} - b^{\prime}_{2} + b_{1}^{\prime } b^{\prime}_{2} G_{14} G_{19} - b_{1}^{\prime } G_{21}$
$G_{23} = c_{2} + c^{\prime}_{2} - c_{1}^{\prime } b^{\prime}_{2} G_{14} G_{19} + c_{1}^{\prime } G_{21}$
$G_{24} = d_{2} + d^{\prime}_{2} - d_{1}^{\prime } b^{\prime}_{2} G_{14} G_{19} + d_{1}^{\prime } G_{21}$
$G_{25} = e_{2} + e^{\prime}_{2} - e_{1}^{\prime } b^{\prime}_{2} G_{14} G_{19} + e_{1}^{\prime } G_{21}$
${\text{G}}_{26} = \tau_{2} \varepsilon_{q2} \varepsilon_{a2} + \varepsilon_{s2} \varepsilon_{q2} \varepsilon_{a2} + G_{22} \tau_{2}^{{{\mathbf{\prime }}2}} \varepsilon_{s2} \varepsilon_{a2} + G_{24} \tau_{2}^{{\mathbf{\prime }}} \varepsilon_{s2} \varepsilon_{a2} + G_{22} \tau_{2}^{{{\mathbf{\prime \prime }}2}} \varepsilon_{s2} \varepsilon_{q2} + G_{25} \tau_{2}^{{{\mathbf{\prime \prime }}}} \varepsilon_{s2} \varepsilon_{q2}$
${\text{G}}_{27} = G_{22} + \frac{{\varepsilon_{s2} \varepsilon_{a2} (G_{24}^{2} - \tau_{2}^{{{\mathbf{\prime 2}}}} G_{22}^{2} ) + \varepsilon_{s2} \varepsilon_{q2} (G_{25}^{2} - \tau_{2}^{{{\mathbf{\prime \prime 2}}}} G_{22}^{2} ) + \varepsilon_{q2} \varepsilon_{a2} (G_{23}^{2} - \tau_{2}^{2} G_{22}^{2} )}}{{G_{26} }}$

Appendix 3

Proof (Proposition 3)

Since the zero function is concave, we can obtain the optimal value of decision variable by considering the values of the first-order partial derivative as zero:

$$ \begin{aligned} \frac{{\partial \pi_{1} (p_{1} ,s_{1} ,q_{1} ,a_{1} ,p_{2} ,s_{2} ,q_{2} ,a_{2} )}}{{\partial p_{1} }} = & \alpha_{1} - 2(b_{1} + b^{\prime}_{1} )p_{1} + b^{\prime}_{1} p_{2} + G_{1} s_{1} - c^{\prime}_{1} s_{2} + G_{3} q_{1} - d^{\prime}_{1} q_{2} \\ & + G_{5} a_{1} - e^{\prime}_{1} a_{2} + (w_{1} - f_{1} )(b^{\prime}_{1} + b_{1} ) = 0 \\ \end{aligned} $$

(38)

$$ \begin{aligned} \frac{{\partial \pi_{1} (p_{1} ,s_{1} ,q_{1} ,a_{1} ,p_{2} ,s_{2} ,q_{2} ,a_{2} )}}{{\partial s_{1} }} = & - \tau_{1} \alpha_{1} + G_{1} p_{1} - \tau_{1} b^{\prime}_{1} p_{2} - (2\tau_{1} c_{1} + 2\tau_{1} c^{\prime}_{1} - \varepsilon_{s1} )s_{1} + \tau_{1} c^{\prime}_{1} s_{2} \\ & - (d_{1} \tau_{1} + d^{\prime}_{1} \tau_{1} + c_{1} \tau^{\prime}_{1} + c^{\prime}_{1} \tau^{\prime}_{1} )q_{1} - \tau_{1} d^{\prime}_{1} q_{2} - (\tau_{1} e_{1} + \tau_{1} e^{\prime}_{1} + c_{1} \tau^{\prime\prime}_{1} + c^{\prime}_{1} \tau^{\prime\prime}_{1} )a_{1} + \tau_{1} e^{\prime}_{1} a_{2} \\ & - (w_{1} + f_{1} )(c_{1} + c^{\prime}_{1} ) = 0 \, \\ \end{aligned} $$

(39)

$$ \begin{aligned} \frac{{\partial \pi_{1} (p_{1} ,s_{1} ,q_{1} ,a_{1} ,p_{2} ,s_{2} ,q_{2} ,a_{2} )}}{{\partial q_{1} }} = & - \tau^{\prime}_{1} \alpha_{1} + G_{3} p_{1} - \tau^{\prime}_{1} b^{\prime}_{1} p_{2} - (\tau^{\prime}_{1} c_{1} + \tau^{\prime}_{1} c^{\prime}_{1} + d_{1} \tau_{1} + d^{\prime}_{1} \tau_{1} )s_{1} \\ & + \tau^{\prime}_{1} c^{\prime}_{1} s_{2} - (2d_{1} \tau^{\prime}_{1} + 2d^{\prime}_{1} \tau^{\prime}_{1} - \varepsilon_{q1} )q_{1} - \tau^{\prime}_{1} d^{\prime}_{1} q_{2} - (\tau^{\prime}_{1} e_{1} + \tau^{\prime}_{1} e^{\prime}_{1} + d_{1} \tau^{\prime\prime}_{1} + d^{\prime}_{1} \tau^{\prime\prime}_{1} )a_{1} + \tau^{\prime}_{1} e^{\prime}_{1} a_{2} \\ & - (w_{1} + f_{1} )(d_{1} + d^{\prime}_{1} ) = 0 \, \\ \end{aligned} $$

(40)

$$ \begin{aligned} \frac{{\partial \pi_{1} (p_{1} ,s_{1} ,q_{1} ,a_{1} ,p_{2} ,s_{2} ,q_{2} ,a_{2} )}}{{\partial a_{1} }} = & - \tau_{1}^{{{\mathbf{\prime \prime }}}} \alpha_{1} + G_{5} p_{1} - \tau_{1}^{{{\mathbf{\prime \prime }}}} b_{1}^{{\mathbf{\prime }}} p_{2} - (\tau_{1}^{{{\mathbf{\prime \prime }}}} c_{1} + \tau_{1}^{{{\mathbf{\prime \prime }}}} c^{\prime}_{1} + e_{1} \tau_{1} + e_{1}^{{\mathbf{\prime }}} \tau_{1} )s_{1} + \\ & + \tau_{1}^{{\mathbf{\prime }}} c_{1}^{{\mathbf{\prime }}} s_{2} - (\tau_{1}^{{\mathbf{\prime }}} e_{1} + \tau_{1}^{{\mathbf{\prime }}} e_{1}^{{\mathbf{\prime }}} + d_{1} \tau_{1}^{{{\mathbf{\prime \prime }}}} + d_{1}^{{\mathbf{\prime }}} \tau_{1}^{{{\mathbf{\prime \prime }}}} )q_{1} + \tau_{1}^{{{\mathbf{\prime \prime }}}} d_{1}^{{\mathbf{\prime }}} q_{2} - (2e_{1} \tau_{1}^{{{\mathbf{\prime \prime }}}} + 2e_{1}^{{\mathbf{\prime }}} \tau_{1}^{{{\mathbf{\prime \prime }}}} - \varepsilon_{a1} )a_{1} + \tau_{1}^{{{\mathbf{\prime \prime }}}} e_{1}^{{\mathbf{\prime }}} a_{2} - \\ & - (w_{1} + f_{1} )(e_{1} + e_{1}^{{\mathbf{\prime }}} ) = 0 \, \\ \end{aligned} $$

(41)

$$ \begin{aligned} \frac{{\partial \pi_{2} (p_{2} ,s_{2} ,q_{2} ,a_{2} ,p_{1} ,s_{1} ,q_{1} ,a_{1} )}}{{\partial p_{2} }} = & \alpha_{2} - 2(b_{2} + b^{\prime}_{2} )p_{2} + b^{\prime}_{2} p_{1} + G_{7} s_{2} - c^{\prime}_{2} s_{2} + G_{9} q_{2} - d^{\prime}_{2} q_{1} + \\ & + G_{11} a_{2} - e^{\prime}_{1} a_{1} + (w_{2} - f_{2} )(b^{\prime}_{2} + b_{2} ) = 0 \, \\ \end{aligned} $$

(42)

$$ \begin{aligned} \frac{{\partial \pi_{2} (p_{2} ,s_{2} ,q_{2} ,a_{2} ,p_{1} ,s_{1} ,q_{1} ,a_{1} )}}{{\partial s_{2} }} = & - \tau_{2} \alpha_{2} + G_{7} p_{2} - \tau_{2} b^{\prime}_{2} p_{1} - (2\tau_{2} c_{2} + 2\tau_{2} c^{\prime}_{2} - \varepsilon_{s2} )s_{2} + \tau_{2} c^{\prime}_{2} s_{1} \\ & - (d_{2} \tau_{2} + d^{\prime}_{2} \tau_{2} + c_{2} \tau^{\prime}_{2} + c^{\prime}_{2} \tau^{\prime}_{2} )q_{2} - \tau d^{\prime}_{2} q_{1} - (\tau_{2} e_{2} + \tau_{2} e^{\prime}_{2} + c_{2} \tau^{\prime\prime}_{2} + c^{\prime}_{2} \tau^{\prime\prime}_{2} )a_{2} + \tau_{2} e^{\prime}_{2} a_{1} \\ & - (w_{2} + f_{2} )(c_{2} + c^{\prime}_{2} ) = 0 \, \\ \end{aligned} $$

(43)

$$ \begin{aligned} \frac{{\partial \pi_{2} (p_{2} ,s_{2} ,q_{2} ,a_{2} ,p_{1} ,s_{1} ,q_{1} ,a_{1} )}}{{\partial q_{2} }} = & - \tau^{\prime}_{2} \alpha_{2} + G_{9} p_{2} - \tau^{\prime}_{2} b^{\prime}_{2} p_{1} - (\tau^{\prime}_{2} c_{2} + \tau^{\prime}_{2} c^{\prime}_{2} + d_{2} \tau_{2} + d^{\prime}_{2} \tau_{2} )s_{2} \\ & + \tau^{\prime}_{2} c^{\prime}_{2} s_{1} - (2d_{2} \tau^{\prime}_{2} + 2d^{\prime}_{2} \tau^{\prime}_{2} - \varepsilon_{q2} )q_{2} - \tau^{\prime}_{2} d^{\prime}_{2} q_{1} - (\tau^{\prime}_{2} e_{2} + \tau^{\prime}_{2} e^{\prime}_{2} + d_{2} \tau^{\prime\prime}_{2} + d^{\prime}_{2} \tau^{\prime\prime}_{2} )a_{2} + \tau^{\prime}_{2} e^{\prime}_{2} a_{1} \\ & - (w_{2} + f_{2} )(d_{2} + d^{\prime}_{2} ) = 0 \, \\ \end{aligned} $$

(44)

$$ \begin{aligned} \frac{{\partial \pi_{2} (p_{2} ,s_{2} ,q_{2} ,a_{2} ,p_{1} ,s_{1} ,q_{1} ,a_{1} )}}{{\partial a_{2} }} = & - \tau_{2}^{{{\mathbf{\prime \prime }}}} \alpha_{2} + G_{11} p_{2} - \tau_{2}^{{{\mathbf{\prime \prime }}}} b_{2}^{{\mathbf{\prime }}} p_{1} - (\tau_{2}^{{{\mathbf{\prime \prime }}}} \, c_{2} + \tau_{2}^{{{\mathbf{\prime \prime }}}} c_{2}^{{\mathbf{\prime }}} + e_{2} \tau_{2} + e_{2}^{{\mathbf{\prime }}} \tau_{2} )s_{2} \\ & + \tau_{2}^{{\mathbf{\prime }}} c_{2}^{{\mathbf{\prime }}} s_{1} - (\tau_{2}^{{\mathbf{\prime }}} e_{2} + \tau_{2}^{{\mathbf{\prime }}} e_{2}^{{\mathbf{\prime }}} + d_{2} \tau_{2}^{{{\mathbf{\prime \prime }}}} + d_{2}^{{\mathbf{\prime }}} \tau_{2}^{{{\mathbf{\prime \prime }}}} )q_{2} + \tau_{2}^{{{\mathbf{\prime \prime }}}} d_{2}^{{\mathbf{\prime }}} q_{1} - (2e_{2} \tau_{2}^{{{\mathbf{\prime \prime }}}} + 2e_{2}^{{\mathbf{\prime }}} \tau_{2}^{{{\mathbf{\prime \prime }}}} - \varepsilon_{a2} )a_{2} + \tau_{2}^{{{\mathbf{\prime \prime }}}} e_{2}^{{\mathbf{\prime }}} a_{1} \\ & - (w_{2} + f_{2} )(e_{2} + e_{2}^{{\mathbf{\prime }}} ) = 0 \, \\ \end{aligned} $$

(45)

Appendix 4

Proof (Proposition 4) First the optimal values of the marketing mix of the follower supply chain (here the first supply chain) are obtained by setting Eqs. (38)–(41) to zero:

$$ p_{1} = ((w_{1} + f_{1} )G_{13} + b_{1}^{\prime } G_{14} p_{2} - c_{1}^{\prime } G_{14} s_{2} - d_{1}^{\prime } G_{14} q_{2} - e_{1}^{\prime } G_{14} a_{2} )G_{19} $$

(50)

$$ s_{1} = \frac{{\varepsilon_{q1} \varepsilon_{a1} G_{2} (p_{1} - w_{1} - f_{1} )}}{{ - G_{14} }} \, $$

(51)

$$ q_{1} = \frac{{\varepsilon_{s1} \varepsilon_{a1} G_{4} (p_{1} - w_{1} - f_{1} )}}{{ - G_{14} }} $$

(52)

$$ a_{1} = \frac{{\varepsilon_{s1} \varepsilon_{q1} G_{6} (p_{1} - w_{1} - f_{1} )}}{{ - G_{14} }} \, $$

(53)

Then, by substituting the optimal values of the second supply chain in the demand and profit function of the supply chain leader (here the first supply chain), we will have:

$$ D_{2} (p_{2} ,s_{2} ,q_{2} ,a_{2} ) = G_{22} p_{2} + G_{23} s_{2} + G_{24} q_{2} + G_{25} a_{2} + (w_{1} + f_{1} ) \, (\frac{{G_{21} }}{{G_{19} G_{14} }} + \frac{{b^{\prime}_{2} G_{13} G_{19} - G_{13} G_{21} }}{{G_{14} }}) \, $$

(54)

$$ \begin{aligned} \pi_{2} (p_{2} ,s_{2} ,q_{2} ,a_{2} ,p_{1} ,s_{1} ,q_{1} ,a_{1} ) = & (p_{2} - w_{2} - f_{2} - \tau_{2} s_{2} - \tau^{\prime}_{2} q_{2} - \tau^{\prime\prime}_{2} a_{2} )\left[ {G_{22} p_{2} + G_{23} s_{2} + G_{24} q_{2} } \right. \\ & { + }G_{25} a_{2} + (w_{1} + f_{1} ) \, (\frac{{G_{21} }}{{G_{19} G_{14} }} + \frac{{b^{\prime}_{2} G_{13} G_{19} - G_{13} G_{21} }}{{G_{14} }})\left. {} \right] - \varepsilon_{s2} \frac{{s_{2}^{2} }}{2} - \varepsilon_{q2} \frac{{q_{2}^{2} }}{2} - \varepsilon_{a2} \frac{{a_{2}^{2} }}{2} - o_{2} \, \\ \end{aligned} $$

(55)

Next, the first order partial derivative of the supply chain leader $\pi_{2}$ in relation to $p_{2} ,s_{2} ,q_{2} ,a_{2}$ is calculated by:

$$ \frac{{\partial \pi_{2} (p_{2} ,s_{2} ,q_{2} ,a_{2} ,p_{1} ,s_{1} ,q_{1} ,a_{1} )}}{{\partial p_{2} }} = D_{2} + G_{22} (p_{2} - w_{2} - f_{2} - \tau_{2} s_{2} - \tau^{\prime}_{2} q_{2} - \tau^{\prime\prime}_{2} a_{2} ) = 0 \, $$

(56)

$$ \frac{{\partial \pi_{2} (p_{2} ,s_{2} ,q_{2} ,a_{2} ,p_{1} ,s_{1} ,q_{1} ,a_{1} )}}{{\partial s_{2} }} = - \tau_{2} D_{2} + G_{23} (p_{2} - w_{2} - f_{2} - \tau_{2} s_{2} - \tau^{\prime}_{2} q_{2} - \tau^{\prime\prime}_{2} a_{2} ) - \varepsilon_{s2} s_{2} = 0 $$

(57)

$$ \frac{{\partial \pi_{2} (p_{2} ,s_{2} ,q_{2} ,a_{2} ,p_{1} ,s_{1} ,q_{1} ,a_{1} )}}{{\partial q_{2} }} = - \tau_{2}^{\prime } D_{2} + G_{24} (p_{2} - w_{2} - f_{2} - \tau_{2} s_{2} - \tau^{\prime}_{2} q_{2} - \tau^{\prime\prime}_{2} a_{2} ) - \varepsilon_{q2} q_{2} = 0 $$

(58)

$$ \frac{{\partial \pi_{2} (p_{2} ,s_{2} ,q_{2} ,a_{2} ,p_{1} ,s_{1} ,q_{1} ,a_{1} )}}{{\partial a_{2} }} = - \tau_{2}^{{{\mathbf{\prime \prime }}}} D_{2} + G_{25} (p_{2} - w_{2} - f_{2} - \tau_{2} s_{2} - \tau_{2}^{{\mathbf{\prime }}} q_{2} - \tau_{2}^{{{\mathbf{\prime \prime }}}} a_{2} ) - \varepsilon_{a2} a_{2} = 0 \, $$

(59)